Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Abstract

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h 0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h 0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h 0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h 0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h 0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.

Aggregate Implications of Firm Heterogeneity: A Nonparametric Analysis of Monopolistic Competition Trade Models

“…Alternatively, the functional form in Assumption 4 can be seen as a functional basis for the nonparametric estimation of ρ g (n) and g (n). Under this interpretation, our estimator is the sieve nonparametric instrumental variable (NPIV) estimator in Chen and Qiu (2016), Christensen (2018), andCompiani (2019). In this case, identification requires the assumption of completeness in Newey and Powell (2003) or, in the case of our model with a linear component, the weaker version of this assumption in Florens et al (2012).…”

“…In this section, we investigate the robustness of the baseline estimates of (n) and ρ(n) presented in Section 5.4. First, we implement the inference procedure in Chen and Christensen (2018) for the estimation of NPIV sieve estimators. Second, we show that results are similar when we use data for different years that have a similar country coverage.…”

“…In this section, we investigate the robustness of our standard estimates of (n) and ρ(n). We estimate standard errors with a criterion value estimated using the bootstrapped procedure in Chen and Christensen (2018). This method accounts for the fact that the function basis used in estimation are intended to approximate the true nonparametric functions (n) and ρ(n).…”

Inference in Nonparametric Series Estimation With Specification Searches for the Number of Series Terms

“…We investigate the following three functions for g(x): g 1 (x) = ln(|6x − 3| + 1)sgn(x − 1/2), g 2 (x) = sin(7πx/2) 1+2x 2 (sgn(x)+1) , and g 3 (x) = x − 1/2 + 5φ(10(x − 1/2)), where φ(•) is the standard normal probability density function, and sgn(•) is the sign function. g 1 (x) is used in Newey and Powell (2003), as well as Chen and Christensen (2018). g 2 (x) and g 3 (x) are rescaled versions used in Hall and Horowitz (2013).…”

Robust inference in deconvolution

“…It is not surprising that such an inference method has been missing for long in the literature, given the technical difficulties of the problem. Deconvolution is an ill‐posed inverse problem, and inference under this problem is known to be challenging; see Bissantz et al (2007), Bissantz and Holzmann (2008), Lounici and Nickl (2011), Horowitz and Lee (2012), Hall and Horowitz (2013), Schmidt‐Hieber, Munk, and Dümbgen (2013), Chen and Christensen (2018), Kato and Sasaki (2018), Babii (2019), Kato and Sasaki (2019), Adusumilli et al (2020) for existing papers developing confidence bands in ill‐posed inverse problems for example. We take a robust inference approach à la Anderson and Rubin (1949), and directly work with the moment restrictions based on Kotlarski's identity.…”